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240 IX ANALYTIC FUNCTIONS

(9.14.2) Under the same assumptions as in (9.14.1), there exists a power

series g^z) = ]T c_nzⁿ, convergent for \z\ < r_l9 and a power series in 1/z and

without constant term, g₂(z) = £ d_nz~ⁿ, convergent for \z\ >r₀, such that

/(z) = gi(z) + 92(2) iⁿ S (" Laurent series" of/). Moreover the power series
g_ly g₂ having these properties are unique, and, for every circuit y in S, we have

1
A »W^c»~2rci

By (9.9.4) we have

1 f f(x)dx » „ _r , . ., If /(*)<&

— £^J—₌y_CnZ» f_or |z| < r_t with c_n = — ₁ ,

27cfj_n x-z _n=₀ ^27rfJ7i ^x

the series being convergent for |z| <r^ On the other hand, for |z| > r₀,
|x| == r₀, we have

1 « x'¹"¹

J n
Z — A _M=i Z

where the right-hand side is normally convergent for \x\ = r₀ (z fixed); by
(8.7.9), we get

— f

2mj_y

with d_n = — xⁿ~ ^lf(x) dx,
ⁿ ^M ^;

x-z " ⁿ 2ni_JQ

the series being convergent for |z| > r₀ . This proves the first part of (9.14.2).
Suppose next we have in S

(9.14.2.1) /(z)=Ia.z" + f;&.z-"

n=0 n=l

both series being convergent in S; let first y be a circuit in S, defined in I;
there are points t, t' in I such that y(t) = inf y(s) = r and y(t') = sup y(s) = r'

sel sel

(3.17.10), hence r₀ < r ^ y(s) ^ r' < r_l for any s e I. But, for r < |z| < r^f,
both series in (9,14.2.1) are normally convergent (9.1.2), hence by (8.7.9),
for any positive or negative integer m

z) dz

= f a_n \ zⁿ⁺™-¹ dz + ^b_n f z-»- ¹ dz.

n=0 Jy M~l J_y

As z^fc+1/(^ + 1) is a primitive of z^k for k ^ -1, we have f z^fe rfz = 0 for

Jer

any circuit cr; (9.14.2) then follows from the definition of the index.

If now y is in S, we remark that there is an open ring S₁:
(1 — e)r₀ < |z| <(1 + s)r₁ contained in A (3.17.11), and we are back to
the preceding case.



240 IX ANALYTIC FUNCTIONS (9.14.2) Under the same assumptions as in (9.14.1), there exists a power oo series g^z) = ]T c_nzⁿ, convergent for \z\ < r_l9 and a power series in 1/z and 00 without constant term, g₂(z) = £ d_nz~ⁿ, convergent for \z\ >r₀, such that /(z) = gi(z) + 92(2) iⁿ S (" Laurent series" of/). Moreover the power series g_ly g₂ having these properties are unique, and, for every circuit y in S, we have 1 A »W^c»~2rci By (9.9.4) we have 1 f f(x)dx » „ _r , . ., If /()<& — £^J—₌y_CnZ» f_or \|z\| < r_t with c_n = — ₁ , 27cfj_n x-z _n=₀ ^27rfJ7i ^x the series being convergent for \|z\| <r^* On the other hand, for \|z\| > r₀, \|x\| == r₀, we have 1 « x'¹"¹

J n Z — A _M=i Z

where the right-hand side is normally convergent for \x\ = r₀ (z fixed); by (8.7.9), we get

— f 2mj_y	_n		with d_n = — xⁿ~ ^lf(x) dx, ⁿ ^M ^;
— f 2mj_y	x-z " ⁿ 2ni_JQ

the series being convergent for \|z\| > r₀ . This proves the first part of (9.14.2). Suppose next we have in S (9.14.2.1) /(z)=Ia.z" + f;&.z-" n=0 n=l both series being convergent in S; let first y be a circuit in S, defined in I; there are points t, t' in I such that y(t) = inf y(s) = r and y(t') = sup y(s) = r' sel sel (3.17.10), hence r₀ < r ^ y(s) ^ r' < r_l for any s e I. But, for r < \|z\| < r^f, both series in (9,14.2.1) are normally convergent (9.1.2), hence by (8.7.9), for any positive or negative integer m

z) dz		= f a_n \ zⁿ⁺™-¹ dz + ^b_n f z-»- ¹ dz. n=0 Jy M~l J_y

As z^fc+1/(^ + 1) is a primitive of z^k for k ^ -1, we have f z^fe rfz = 0 for Jer any circuit cr; (9.14.2) then follows from the definition of the index. If now y is in S, we remark that there is an open ring S₁: (1 — e)r₀ < \|z\| <(1 + s)r₁ contained in A (3.17.11), and we are back to the preceding case.